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A 'doodle' is a closed intersecting curve drawn without taking
pencil from paper. Only two lines cross at each intersection or
vertex (never 3), that is the vertex points must be 'double points'
not 'triple points'. Number the vertex points in any order.
Starting at any point on the doodle, trace it until you get back to
where you started. Write down the numbers of the vertices as you
pass through them. So you have a [not necessarily unique] list of
numbers for each doodle. Prove that 1)each vertex number in a list
occurs twice. [easy!] 2)between each pair of vertex numbers in a
list there are an even number of other numbers [hard!]

How many different cubes can be painted with three blue faces and
three red faces? A boy (using blue) and a girl (using red) paint
the faces of a cube in turn so that the six faces are painted in
order 'blue then red then blue then red then blue then red'. Having
finished one cube, they begin to paint the next one. Prove that the
girl can choose the faces she paints so as to make the second cube
the same as the first.

A player has probability 0.4 of winning a single game. What is his
probability of winning a 'best of 15 games' tournament?

One Basket or Group Photo

Stage: 2, 3, 4 and 5 Challenge Level:

A school photographer is taking a photograph of the two basketball
teams. She has to arrange ten people, all of different heights, in
two rows of five, one behind the other. Each person at the back
must be taller than the person directly in front of them. Along the
rows the heights must increase from left to right.

In how many ways can two, four or six people to be arranged in
this way for a photo, or eight people? In how many ways can the ten
team members be arranged like this for the photo to be taken?

You may even like to generalise the problem to twelve people or
to any specified even number.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities
can be found here.